Proving a left identity

Question

I have the following question:

I know that I need to prove that any element with a+b=1 multiplied by S will yield S. Can anyone give me a starting point for this proof?

Answer 1

An element with $a+b=1$ is a matrix $$\begin {bmatrix} x & 1-x\\x & 1-x\end {bmatrix} $$

Upon left multiplication by $$\begin {bmatrix} a & b\\a & b\end {bmatrix} $$ we get $$\begin {bmatrix} x & 1-x\\x & 1-x\end {bmatrix} \begin {bmatrix} a & b\\a & b\end {bmatrix} =\begin {bmatrix} a & b\\a & b\end {bmatrix}$$

Thus the statement is valid.

Answer 2

Hint: Multiply $\pmatrix {x&1-x\\ x&1-x}$ with $\pmatrix {a&b\\ a&b}$.

Answer 3

$$ \begin{bmatrix} a&b\\a&b\end{bmatrix}\cdot\begin{bmatrix} c&d\\c&d\end{bmatrix}= \begin{bmatrix} c&d\\c&d\end{bmatrix} $$ for any $c$, $d$, because $a+b=1$. Can you check this?